Computing distance as a function of time

[tony873004]

I want to compute the position of a planet from its star as a function of time.
Here is an illustration describing the problem: http://orbitsimulator.com/PF/pft.GIF

Each of these 5 planets has a semi-major axis of 1 AU and a period of 1 year. The case of the planet in a circular orbit is easy. It's distance doesn't change, so for this planet, d(t)=sma, where sma is the semi-major axis of the planet.

But the other planets have eccentric orbits, causing their distances to vary. They travel fastest when near the star.

I made a plot of their distances vs time using a numerical method. But I'd like to what analytic formula could give me these distances as well. Here's the graph: http://orbitsimulator.com/PF/pft2.GIF
The y-axis is in meters, the x-axis in days. The names of the lines reveal the eccentricity of the planet (p8 is 0.8).

The graph for p2 looks like a sin wave, but the higher the eccentricity, the pointier the bottom of the sin wave. Is there a formula to describe these curves?

Thanks!

 

[DaveC426913]

I'm imagining converting them to polar coordinates might yield some insight.

 

[tacman]

Thank you for sharing your problem and the accompanying illustrations. Computing distance as a function of time is a common problem in the field of astronomy and astrophysics. In order to accurately determine the position of a planet from its star as a function of time, we need to take into account the planet's orbital characteristics, such as its semi-major axis and eccentricity.

As you mentioned, for a planet in a circular orbit, the distance remains constant and can be easily calculated using the formula d(t) = sma, where sma is the semi-major axis of the planet. However, for planets with eccentric orbits, the distance varies as the planet moves closer to and further away from the star during its orbital period.

To determine the distance of a planet from its star at any given time, we can use the formula for Kepler's second law, which states that the area swept out by the line connecting the planet to the star is equal for equal times. This can be expressed as:

1/2 * r^2 * d(theta)/dt = A

where r is the distance from the planet to the star, d(theta)/dt is the angular velocity of the planet, and A is the area swept out by the line connecting the planet to the star.

Using this formula, we can derive an equation for the distance of a planet from its star as a function of time, taking into account the planet's eccentricity. This equation is:

r(t) = sma * (1 - e^2) / (1 + e * cos(theta(t)))

where sma is the semi-major axis, e is the eccentricity, and theta(t) is the angle between the line connecting the planet to the star and the major axis of the planet's orbit at time t.

Using this formula, we can plot the distance of a planet from its star as a function of time, as shown in your graph. As you observed, the graph for a planet with higher eccentricity has a pointier bottom, reflecting the fact that the planet moves faster when it is closer to the star.

In conclusion, there is indeed an analytic formula to describe the distance of a planet from its star as a function of time, taking into account its orbital characteristics. This formula can be derived from Kepler's second law and can be used to accurately determine the position of a planet from its star at any given time. I hope this helps to answer your question and provides a solution to your problem.